3. Consider the vector field--(z+92)--Idzy2 + 412zk. Calculate / E.dr from (0,0,0) to (1.1,1) along (a) r f2 [10 marks] 0,0,0) to (b) the path from (0,0,0) to (1) consisting of the straight lines fro...
(1) Integrate f(x, y,z)+Vy - z2 over the straight line segment path from (0,0,0) to (1,1,1) (2) Consider the field F (2xyz+2,x2z, x2y), (a) (b) (c) Show that the field is conservative. Find a potential function for the field. Find the work the field does on an object that follows the path consisting of the line segment from (0,0,0) to (1,2,2), followed by the line segment from (1,2,2) to (2,4,3) Find the work done by the field ß-(x, 3y,-5z) along...
1. Given force F-xi 4zj Path 1: (0,0,0) to (0,1,1) Path 2: (0,1,1) to (1,1,0) Path 3: (1,1,0) to (0,0,0) 5yk acts following the path: . Sketch the motion in Cartesian coordinate Write the parametric equation for each path Calculate the total work done by force F which is moving from Path 1 to 3 2. A vector M is defined by M-x2yi xyj and region R corresponds to Sketch the region of R in Cartesian coordinate . Evaluate the...
please don't solve anything. by mistake this question 1 Verify Cauchy's theorem for the closed path c consisting of three straight lines joining A (1+1), B (34 j3), C (-1 + β) where r(z) = z-1 + j. mistnigof threstraign 79 the ueniane under the transformation w = f(z We were unable to transcribe this image 1 Verify Cauchy's theorem for the closed path c consisting of three straight lines joining A (1+1), B (34 j3), C (-1 + β)...
Problem #7: Let R = r \ {(0,0,0)) and F is a vector field defined on R satisfying curl(F) = 0. Which of the following statements are correct? [2 marks] (1) All vector fields on R are conservative. (ii) All vector fields on Rare not conservative. (iii) There exists a differentiable function / such that F - Vf. (iv) The line integral of Falong any path which goes from (1,1,1) to (-2,3,-5) and does not pass through the origin, yields...
Please make it simple and clear to understand 3. A vector field is given by (a) Show that the vector field r is conservative. Then find a scalar potential function f(r,y,) such that r - gradf and f(0,0,0) 0 (b) By the result of (a) the following line integral is path independent. Using the scalar potential obtained in (a) evaluate the integral from (0,0,2) (where-y-0) to (4,2,3) (where -1,y 0,2) 4.2,3) J(0,0,2) 3. A vector field is given by (a)...
q4 please thanks (1) Let A - (0,0), B- (1,1) and consider the veetor field f(r, y,z)vi+aj. Evaluate the line integral J f.dr )along the parabola y from A to B and (i)along the straight line from A to B. Is the vector field f conservative? (2) For the vector feld f # 22(r1+ gd) + (x2 + y2)k use the definition of line integral to (3) You are given that the vector field f in Q2 is conservative. Find...
Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the point (1,1) in terms of and y (b) Let u R> R3 be a C3 path parametrised in terms of t. Evaluate and simplify d dt Question 3 (11 marks) (a) Consider the vector field F(r, y)yaj (i) Determine V'F. (ii Determine the equation for the flow line of F passing through the...
#3 Consider the vector field F- Mi+ Nj Pk defined by: F- ysinzi+sinjry cos z k. Compute the line integral ScF dr over a unit circle. Compute the line integral ysin z dr+ r sin z dy + ry cos zdz (0,0,0) #3 Use Green's Theorem to evaluate the line integral along the given positively orientated curve C. e2*t d e" dy, C is the triangle with vertices (0,0), (1,0), and (1,1) #3 Consider the vector field F- Mi+ Nj...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...